By Gerard A. Maugin
Exploring fresh advancements in continuum mechanics, Configurational Forces: Thermomechanics, Physics, arithmetic, and Numerics provides the overall framework for configurational forces. It additionally covers a number of functions in engineering and condensed subject physics. the writer offers the basics of permitted general continuum mechanics, ahead of introducing Eshelby fabric pressure, box conception, variational formulations, Noether’s theorem, and the ensuing conservation legislation. within the bankruptcy on complicated continua, he compares the classical point of view of B.D. Coleman and W. Noll with the point of view associated with summary box concept. He then describes the real idea of neighborhood structural rearrangement and its dating to Eshelby pressure. After the relevance of Eshelby pressure within the thermodynamic description of singular interfaces, the textual content specializes in fracture difficulties, microstructured media, structures with mass exchanges, and electromagnetic deformable media. The concluding chapters speak about the exploitation of the canonical conservation legislation of momentum in nonlinear wave propagation, the appliance of canonical-momentum conservation legislations and fabric strength in numerical schemes, and similarities of fluid mechanics and aerodynamics. Written through a long-time researcher in mechanical engineering, this ebook offers a close remedy of the speculation of configurational forces—one of the most recent and so much fruitful advances in macroscopic box theories. via many purposes, it exhibits the intensity and potency of this concept.
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Extra info for Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics
1). 2) this is called a conservation law, but here it is in fact a nonconservation because of the lack of invariance of the physical system under material translation. 4) reduces to a strict conservation law. 7) which may be referred to as the Ericksen identity of elasticity, as Ericksen (1977) gave it in the special case (no body force, no elastic inhomogeneity): (divR T ). F + (divR b) = 0. 2), these are none other than special cases of the celebrated Noether’s identity. R emar k 3 . 138), we now understand why the latter was referred to as the Eshelby format of the Cauchy stress.
F −1 , σ = −2 F − T . 134) while the second and first Piola–Kirchhoff stresses are given by . C −1 , T = − J F −1 . F −1 . 135) The proof of these identities exploits the following relations (written in components to facilitate the computations): ∂( F −1 ) j L ∂F i K = − ( F − 1 )i ( F − 1 ) j L K KM (C−1 )LN . 137) We remark that w = J F−1W is the free energy per unit volume in the actual configuration. Then we ask the reader to check that the Cauchy stress may be given the following form (Maugin and Trimarco, 1993): σ = w1 − F − T .
Prove the objectivity of the field DJA. As a matter of fact, define the following material vector by means of a transport by rotation (instead of = RT A and prove that standard pull back), A E x ercise: ∂A DJ A = R − T . 58) Generalization of the definition of convective and Jaumann derivatives to tensors of any order is straightforward. R emar k 2 . 60) Ω = ω or Ω ij = ω ij = u [i , j] . 62) where η is an infinitesimal parameter and ω = –ωT is finite. 62 introduces the notion of infinitesimal generator of the rotation (orthogonal) group O(3).